Problem 12 - Fundamentals of Physics

Question

In an oscillating \(L C\) circuit, when \(75.0 \%\) of the total energy is stored in the inductor's magnetic field, (a) what multiple of the maximum charge is on the capacitor and (b) what multiple of the maximum current is in the inductor?

Step-by-Step Solution

Answer

(a) 0.5 times maximum charge, (b) 0.866 times maximum current.

Step 1: Understand Energy Distribution in LC Circuit

In an LC circuit, energy oscillates between the inductor's magnetic field and the capacitor's electric field. Let \( U \) be total energy. When a certain percentage of this energy is stored in the inductor, the sum of energy in the capacitor's electric field and the inductor's magnetic field equals \( U \).

Step 2: Express Energy in Inductor

The magnetic energy in the inductor, \( U_L \), is \( U_L = \frac{1}{2} L I^2 \), where \( L \) is inductance and \( I \) is the current. Given \( U_L = 0.75 U \), relate this to other circuit parameters.

Step 3: Express Energy in Capacitor

The energy in the capacitor, \( U_C \), is \( U_C = \frac{1}{2} C V^2 \), where \( C \) is capacitance and \( V \) is voltage. When \( U_L = 0.75 U \), then \( U_C = 0.25 U \).

Step 4: Relate Maximum Energy Capacitor to Charge

Using the relationship \( U = \frac{1}{2} C V_{max}^2 \), and charge \( Q = C V \), we find \( U_C = \frac{Q^2}{2C} \). Hence \( Q = \sqrt{0.25} Q_{max} = 0.5 Q_{max} \), with \( Q_{max} \) being maximum charge.

Step 5: Relate Maximum Energy Inductor to Current

Using the relationship \( U = \frac{1}{2} L I_{max}^2 \), \( U_L = \frac{L I^2}{2} \). Hence, \( I = \/sqrt{0.75} I_{max} \approx 0.866 I_{max} \), with \( I_{max} \) being the maximum current.

Key Concepts

Energy Distribution

In an LC circuit, energy is continuously exchanged between two main components: the inductor and the capacitor. During this process, the energy oscillates back and forth from the magnetic field of the inductor to the electric field of the capacitor. At any given moment, the total energy in the circuit, denoted as \( U \), is split between these two fields.
For example, if 75% of the total energy is stored in the inductor's magnetic field, the remaining 25% is then in the capacitor's electric field. The equation reflecting this distribution is \( U = U_L + U_C \), where \( U_L \) is the energy stored in the inductor, and \( U_C \) is the energy stored in the capacitor. Understanding this interchange of energy is crucial in analyzing and predicting the behavior of LC circuits.
When working with LC circuits, remember:

• Energy in the inductor is described by its magnetic field.
• Energy in the capacitor is stored as electric potential.
• The sum of energies in both components equals the total energy \( U \).

Inductor's Magnetic Field

The magnetic field in an inductor in an LC circuit stores energy that is proportional to the square of the current flowing through it. This energy is often referred to as magnetic energy and is given by the formula \( U_L = \frac{1}{2} L I^2 \), where \( L \) is the inductance and \( I \) is the current at a specific instant.
When 75% of the LC circuit's energy is stored in this magnetic field, we apply \( U_L = 0.75 U \). This condition provides insights into how the current at that moment relates to the maximum current, \( I_{max} \).
To use the energy equation effectively:

• Identify the inductance (\( L \)).
• Measure or calculate the current (\( I \)) at the instant of interest.
• Check how the energy relates to the total circuit energy \( U \).

Maximum Charge

The charge stored in the capacitor of an LC circuit is central to its functioning. This charge builds up as energy is transferred from the magnetic field of the inductor to the electric field of the capacitor. The formula for the electrical energy stored in the capacitor is \( U_C = \frac{1}{2} C V^2 \), where \( C \) is the capacitance and \( V \) is the voltage across the capacitor.
When the energy stored in the capacitor equals 25% of the total, we can use the relationship \( U_C = \frac{Q^2}{2C} \) to find the charge \( Q \) at that moment. Solving, we find \( Q = 0.5 Q_{max} \), where \( Q_{max} \) is the maximum charge the capacitor can hold.
Keep in mind:

• The relationship \( Q = C V \) relates charge and voltage.
• The charge at any instant can be determined from the energy stored.
• Maximum charge is key to setting the boundaries of circuit capacity.

Maximum Current

In the oscillation of an LC circuit, the current through the inductor reaches a maximum when energy is entirely stored in the magnetic field of the inductor. The concept of maximum current, \( I_{max} \), is pivotal, as it characterizes the peak energy transfer rate.
When energy distribution indicates that 75% of the circuit's total energy is in the inductor, we can derive the relationship \( I = \sqrt{0.75} I_{max} \) or approximately \( 0.866 I_{max} \). This formula allows us to find the current at the time of energy measurement, in relation to its maximum value.
For practical application:

• Use \( U_L = \frac{1}{2} L I^2 \) to understand energy at any current level.
• Relate the current at any instant to maximum possible current \( I_{max} \).
• Ensure inductors and components can handle peak currents specified by \( I_{max} \).